Spaced spaces

C ^OMPOSITIO M ^ATHEMATICA

I. M ^OERDIJK

Spaced spaces

Compositio Mathematica, tome 53, n

^o

2 (1984), p. 171-209

<http://www.numdam.org/item?id=CM_1984__53_2_171_0>

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171

SPACED SPACES

I.

Moerdijk

© 1984 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands.

Contents

Introduction ... [1] 171

Part 1. Topological spaces in sheaves ... [3] 173

§1. Representation ^of topological ^spaces ... [3] ¹⁷³

§2. ^The category of internal topological spaces ... [7] ¹⁷⁷

§3. Pseudometric spaces ... [12] ¹⁸²

§4. Pseudometric and metric spaces in sheaves ... [14] ¹⁸⁴

Part 2. Spaces in sheaves and sheaves of spaces ... [17] 187

§1. ^A general adjunction ^theorem ... [17] ¹⁸⁷

§2. Internal _spaces over a locally compact zero-dimensional base space ^... [20] ¹⁹⁰

Part 3. Change of base space ... [22] ¹⁹²

§1. Direct and inverse images ... [22] ¹⁹²

§2. The category of spaced spaces ... [26] ¹⁹⁶

§3. Factorizations in SPSP ... [28] 198

§4. Limits and colimits of spaced spaces ... [31] 201

References ... [38] 208 Introduction

The model

theory

^of intuitionistic

higher

^order

logic,

^as e.g. described in

[1],

^{makes it}

possible

^to

interpret

intuitionistic mathematics in

categories

of sheaves. This has

proved

^{to be}

particularly

^{useful in}

algebra (see [6]).

Here "internal" mathematical

objects correspond

to structures which are

familiar from the

theory

^{of sectional}

representations.

^For

example,

^a

ring (=

commutative

ring

^with

l)-object

^{in the}

category Sh(X)

of Set-valued sheaves on a

topological space X

^{is the same as a sheaf of}

rings on X,

^{or a} sheaf of sets on X with ^a continuous

ring

^{structure on the stalks. In order}

words, doing ring theory internally

^{in the}

category Sh(X)

^{coincides with}

studying ringed

spaces with base space X.

In this paper, ^we will discuss what

happens

^{when one}

replaces "ring"

by "topological space"

in the above: we will deal with the

question

^of

what

topological space-objects

ⁱⁿ

categories

of sheaves look

like,

^{and we}

will consider the relation between such

topological space-objects

^and

sheaves with values in the

category

^of

topological

spaces and continuous functions.

Our

approach

will be rather

category-theoretic:

^{we will} concentrate on

equivalence

^of

categories-theorems (representation theorems),

^{and on}

adjunctions

^between

categories. By doing this,

^we

hope

^to

provide

^some

general background

^to the models for intuitionistic

topology

of the kind discussed

by Grayson [4],

^for

example.

^In intuitionistic

topology (or, topology

ⁱⁿ

sheaves)

^{one finds a lot of}

pathological

situations which are

due to a lack of

points. Many

^{of the}

pathologies disappear

^{when one does}

intuitionistic

topology

^without

points,

^that

is,

^locale

theory.

^{In this} sense, locale

theory

^{seems to be} the proper way of

doing topology

^{when one’s}

underlying logic

^is intuitionistic

logic. Still,

in this paper ^we will not consider

locales,

^but

study

models for intuitionistic

topology

^{in the more}

traditional sense of

[9], [4],

^{where a} space is a set

of points

^{with some}

additional structure.

Let us

briefly

outline how this paper is

organized.

^{We assume the}

reader to be

acquainted

with the basis facts of

general topology

^and

category theory,

^{and to have a}

thorough understanding

of the model

theory

^of intuitionistic

(higher order) logic

^as described in

[1].

This paper

[1]

will be the

starting point

for the first

part,

^{where we} consider external

representations

^of

topological

space

objects

ⁱⁿ

categories

of sheaves. We prove Stout’s

representation theorem,

and derive Fourman’s

representa-

tion theorem for sober spaces ^{as an} easy

corollary.

^In the second

part,

^we will consider the relation between the

category TOP(Sh(X))

^of

topologi-

cal space

objects

ⁱⁿ

Sh(X),

^{and the}

category Sh(X, TOP )

of sheaves ^on X with values in the

category

^of

topological

spaces and continuous functions. We prove ^a

general adjunction theorem,

^{and we} show that in the case of a

locally compact

zero-dimensional

base-space ^X, TOP(Sh(X))

^is

(equivalent to)

^a reflective

subcategory

^of

Sh ( X, TOP ).

In the third

part,

^{we consider}

change

of base space, and

investigate

^some

of the structure of the

category

^of

*’spaced spaces",

which is defined as an

analogon

^{of the}

category

^of

ringed

spaces, ^or the

category

^of

geomet-

ric spaces, familiar from

algebraic geometry.

This paper has

quite

^a

long history.

^A first version was written in the fall of 1980 as

[5].

^{The main reason for the}

delay

ⁱⁿ

producing

^the

present

version ^was that the central

representation

^theorem

(Theorem 5,

^Part

1,

Section 1

below)

^{turned out to have been}

proved independently,

^but

much

earlier, by

^Stout

(cf. [7]).

¹ would like to thank L.N. Stout for

bringing

the existence of

[7]

^to my attention.

Also,

1 would like to thank

professors

^van Dalen and Troelstra for

helpful

^{comments on} the earlier version

just mentioned,

^{and for}

encouraging

^{me to} write the

present

version.

PART 1. TOPOLOGICAL SPACES IN SHEAVES

1.

Representation

^of

topological

spaces

Throughout

this paper, ^we will ^use the well-known

representation

^theo-

rem for sheaves

saying

^{that a sheaf A over a}

topological

space ^T is

representable

^as the sheaf of continuous sections of a local homeomor-

phism EA

^{T. This}

correspondence

^{is an}

equivalence

^of

categories.

T

Natural transformations A - B from one sheaf A over T to another are

f

represented by

continuous maps

EA E.

^over

T ;

^{i.e. T acts on sections}

by just composing

^with

f. (For

^more

details,

^see e.g.

[1], [2].)

^{Given this}

representation,

^we may either think of sheaves ^as Set-valued

functors,

^or

as local

homeomorphisms.

Since in the definition of a sheaf over a space T we need

only

^{refer to} the lattice

fP(T)

of open subsets of

T,

^{we will}

assume that the

base-space

^{T is}

sober,

whenever this is convenient.

If

E 1

^T is ^a local

homeomorphism,

^an open

neighbourhood (nbd) Ue

of a

point

e E E will be called small if

p Ue : Ue ~ p ( Ue )

^{c T is a}

homeomorphism. By definition,

^{the small}

neighbourhoods

^{form a basis}

for E.

Let us now turn to

topological

spaces in sheaves.

Using

^the

interpreta-

tion of

higher-order logic

in sheaves as

presented

ⁱⁿ

[1],

^{we can define a}

topology

^{on a sheaf A over T as a}

subobject O(A) of (A)

^{such that}

as in the classical case. Our aim in this section is to

give

^{an external}

representation

^of

topological space-objects

ⁱⁿ

Sh(T).

1.1. LEMMA: Let A be a

sheaf

^{on T. Then}

global

^elements

of f1lJ(A)

ⁱⁿ

Sh(T) correspond

^to open subsets

of EA,

^with

equality given by

for

open subsets

O, O’ of EA .

PROOF: A

global

^element

of 9lJ(A)

^{is a}

(strict

^and

extensional) predicate

A -

O(T) ([1]).

An open

0 C EA

defines such ^a

predicate Po by PO(a) = a-1(O). (Here

^we

identify

elements of A with sections of the

representing

local

homeomorphism EA ^T). Conversely,

^a

predicate

^{P: A ~}

O(T)

defines ^an open ^set

To prove the

correspondence,

^{we show that}

ad 1.

(~)

Take a ~ A . We have to show that

P(a)~~{a-1(U)|U

^is

small and

P((p~ U)-1)=p(U)}.

^Take

t~P(a),

and let U be a small nbd of

a(t)

^{such that}

p(U)~P(a) ( p

^is

continuous).

^Since

p

^U:

U ~ p(U)

^{is a}

homeomorphism,

^we derive that

ap(U) = (p U)-1,

and hence

P(( p U)-1)

⁼

P(a r p(U))

⁼

P(a) n p (U) = p(U). ( ç )

^Take

a E A and U a small open subset of

EA .

^{We have to} show that if

P((p U)-1)=p(U), then a-1(U)~P(a).

So suppose

P((p U)-1) = p(U).

^Since

a-1(U)=[a=(pU)-1],

^{we obtain}

P(a)~a-1(U) =

P(a[a=(p U)-1])=P((p U)-1[a=(p U)-1])=p(U)~a-1

(U)

⁼

a-1(U).

^Hence

a-1(U)

^c

P(a).

ad 2. We have to show that for an open 0 ^c

EA,

But

(pU)(O) = p(U) iff p(O~ U) = p(U),

^iff

^{U c O,}

^so this is im- mediate from the fact that the small open ^sets form ^a basis. 0

The

following

^{lemma is an}

analog

^of

8.12(i)

^of

[1].

1.2. LEMMA: Let T be an internal

topology

^{on a}

sheaf

^{A over T}

( i. e.

^{T is a}

subsheaf of 9(A) satisfying

^the

definition (*) given above).

^Then every element

of

^{T is the} restriction

of

^a

global

^element

of

^T.

PROOF: As

usual,

^T may also be

regarded

^{as a}

predicate on 8P(A),

^{and for}

O~O(EA)

^{we write}

(O) = [O~].

^{Now let}

(0, U )

^{be a} section of ^T

over U E

O(T),

^{i.e. O E}

O(EA)

^{is a}

global

^element

of 9(A) (cf.

^Lemma

1)

^with

O(a) := a-1(O) ~

U for all

a ~ A,

^and

U ~ O(T)

^{is such that}

U c

(O).

Consider the

predicate

^{u :}

(A) ~ O(T)

^defined

by

(the

last inclusion

by extensionality

^of

T ).

^{Hence since}

by

^definition

1.3. REMARK: Let ^us write r for the

global

sections functor. Then the

proof just given

shows that for any internal

topology

^T

^{on A,}

P

1.4. DEFINITION: Let E - T be a local

homeomorphism.

^A

p-topology

^on

E is a

topology

^{r on E which is coarser then the}

original topology

^{on E}

(i.e.

^{r c}

O(E)),

^{and makes} p continuous

(in

^{the sense that}

p-1(U) ~

^r

for every

U~O(T)).

1.5. THEOREM: Let A be a

sheaf

^over

^T, represented by

the local homeomor-

phism EA

^{T. Then internal}

topologies

^{on A}

correspond

^to

(external) p-topologies

^on

EA.

PROOF: An internal

topology

^{is a}

predicate

^{T on}

9(A) satisfying

^the

definition

( * ) given

^{above. T} is determined

by

its restriction T:

O(EA) ~ f2(T),

^{which must be an} extensional function. Let rT be the set of

global

elements of T. We claim that:

(1)

^{rT is a}

topology

^on

EA

which makes _p continuous. The latter

part

of

(1)

^{follows from} Remark 1.3. As for the first

part, clearly

^{E rT and}

EA ~ 0393. Also,

^if

O,O’ ~ 0393,

then because

(O) ~ (O’) ~ (O~O’),

O n O’ E rT.

Finally,

^{if ~}

^rT,

consider the

(global) predicate

^u:

O(EA)

~ O(T)

^defined

by

By definition, ~O(u(O)~(O))~(~uj).

^But

^Uu corresponds

^{to the}

open subset

U4Y

of

EA (since [a~~u]=~{u(O"0~[a~O’[|O’ ~

O(EA)}

⁼

a-1~)),

and for all 0 ^E

C9(EA), u(O)

ç

(O) (since u(O) =

~{U~O(T)|O~p-1(U)~}~~{U~O(T)|O~p-1(U)~0393}; using

Remark

3, O~p-1(U) ~ 0393 implies U ~ (O),

^so

u(O) ~ (O )).

^Com-

bining these,

^we

get (~u) = T,

^so

U

^E rT.

Conversely,

^{let r c}

(9(E,)

^{be a}

topology

^on

EA making p

continuous.

Define a

predicate Tr : (a) ~ O(T) by setting,

^{for 0 e}

O(EA),

(by

^Remark

^3,

^{we are forced to do}

so).

^Then

(2)

Tr is ^an internal

topology

^on A. To show

(2),

^{first note that}

clearly lL0 E Tpj

⁼

[EA

^E

Tr]

⁼

^T,

^{and for all}

globals O, O’ ~ O(EA), [O ~ 0393]

ⁿ

[O’ ~ 0393] ^{~ [O ~}

^{0’ E}

0393].

^{It is}

slightly

less trivial that _Tr is

internally

closed under unions. Take ^a

global predicate

^u:

O(EA) ~ O(T)

^on

(A);

we have to show that

For sections a of

EAT, [a~~u]=[~O~u·a~ 0]) =~{a-1(O)~

u(O)|O ~ O(EA)} = a-1(~{O ~ p-1u(O)|O ~ O(EA)}).

^{Now suppose}

W~[u(O)~0393(O)]

^{for all}

O~O(EA).

^{We claim that}

W~0393(~u).

W~[u(O)~0393(O)]

^{means that}

W~u(O)~~{U|O~p-1(U)~0393}.

Hence for all

O ~ O(EA), O ~ p-1(W~u(O))~0393 (since

^r

makes p continuous).

^So

But this _says that

by

definition of _Tr. This shows that _Tr is an internal

topology.

To prove the

correspondence,

^{we show that}

(1)

for any internal

topology

^T,

_{Tr =}

^T

(2)

for any

p-topology

^{r on}

EA, Fïp

^{= r.}

(2)

is trivial:

O ~ 03930393~0393(O) = T ~ O~p-1(T)~0393~O~.

^For

⁽¹⁾

note that because r is an internal

topology,

it suffices

by

^{Lemma 1.2 to}

show that

rTrT

=.Fr. But this follows from

(1).

^This

completes

^the

proof

of the theorem. 0

In the next

section,

^{we will} reformulate this

correspondence

^{as an}

equivalence

^of

categories.

^{Let us} first look at two kinds of internal spaces.

g

1.6. EXAMPLES:

(a)

^Let

Y -

^T be a continuous function from a

topologi-

cal _{space Y} to the base _{space T.} The sheaf of sections of _g carries a

natural internal

topology T

^induced

by

^the

topology

^{of Y:}

global

^elements

a

of ^{r are} the

predicates Po,

^{0 e}

C9(Y),

^{where for a} section U - Y of _g,

PO(a):=a-1(O).

^{It is}

readily

checked that this indeed defines a

topology.

g

This internal _space is called the _space

of

^{sections of Y ~}

^T,

^{and is}

denoted

by

gT.

(b)

^{As a}

special

^{case of}

(a),

^{let X be a}

topological

space, and consider the internal space of sections of the

projection X

^{X T T.} This space is denoted

by XT,

and is called the constant space associated with X.

Below,

^{we shall} return to the constructions of internal spaces from external _ones, and

investigate

^their

categorical properties.

2. The

category

of internal

topological

^spaces

We mentioned above that a sheaf A over T can be

represented

^{as a sheaf}

of sections of ^a local

homeomorphism EA ^T,

^{and that}

sheaf-maps (natural transformations)

^{A - B}

correspond

^to continuous functions

EA EB

^{over T.} Now suppose ^we have two sheaves A and B over

T, equipped

with internal

topologies

^{T and a}

respectively.

^An

(internal)

f

continuous function

f:(A,)(B,03C3) is a sheaf-map A - B

^{such that}

1= VO E

(B)(O~ 03C3~f-1(O)~),

^as usual. Let

EA EB

be the repre- sentation of

f.

^Then

(identifying

elements of A and B with sections of

EA ^T, EB ^T, etc)

^{we find the}

following correspondence.

2.1. LEMMA: A

sheaf-map f : ( A, ) ~ (B, a)

^is

internally

continuous in

S’h ( T ) iff

^its

representation f

is continuous w. r. t. rT on

EA

^{and fa on}

EB.

PROOF:

f

^and

f

^{are related}

through f(a)=oa

for all sections a of _p.

Further,

^{for a}

global

^{element 0}

of (B),[a ~f-1(O)]

⁼

[f(a)~ ⁰¹

⁼

a-1(-1(O)).

^Therefore

(using

Remark 3 of Section

1), if f is

^continuous

(w.r.t. ra, f7),

then if 0 is a

global

^element

of .9(B),

so

f

^is

internally

continuous.

Conversely, if f is internally continuous,

^then

(using [a~f-1(O)] = a-1(-1(O))

^as

above)

^{0 E ra}

implies -1(O) ~

^{Fr. D}

Putting

Theorem 5 of the

preceding

section and this lemma

together,

we obtain an

equivalence

^of

categories.

^Let

TOP(Sh(T))

denote the

category

of internal

topological

spaces and

(global)

continuous functions in

Sh ( T ),

^{and let}

SPSP( T ) (" spaced

spaces, with base space

T ")

^denote

the

category

^whose

objects

^are of the form

(E, 0393) T,

where E is ^a

topological

space, p: ^{E ~ T} is a local

homeomorphism,

and r is a

p-topology

^on

^E,

^{and whose arrows from an}

object (E, 0393)

^{T to an}

object (F, 0394)

T are functions

f :

^{E - F over T which are continuous}

w.r.t. both

topologies

^on

E,

^F.

2.2. THEOREM: The

categories TOP(Sh(T))

^and

SPSP(T)

^are

equivalent.

a

Let us return to spaces of sections

(1.6.)

^{for a moment. If we}

apply

^the

representation

to constant spaces

XT

^we

get

that these are

represented by

TT

structures

(XT, O(XT)) ~ ^T,

^where

XT

^{is an external}

topological

space

(note

^{that we use}

XT

^{to denote two different}

things,

^an internal space and

an external

one!), XT ~

^{T a local}

homeomorphism,

^and

O(XT)

^a

03C0-topol-

ogy ^on

XT.

^The external space

XT

^is calculated in the standard way

(cf.

[2]).

^{Let us}

quickly

^{recall some details.}

XT

^{has as its set of}

points

(here [f]t denotes

^the

equivalence-class of f with respect

^{to local}

equality

of functions at

t).

^The

mapping

^03C0:

XT ~ X

^{is defined}

by setting 03C0[f]t, t&#x3E;

^{= t. The}

topology

^on

XT making

^{v a local}

homeomorphism

^has

f

as basic opens the ^sets

[ f , U ],

for U open ^{c T} and U - X

continuous,

where

The

ir-topology O(XT)

^on

XT

^{has as} opens the ^sets

for U on open subset of the

product X

^{X T. It is}

readily

checked that this is indeed ^a

’1T- topology.

p

Now internal continuous functions from a space

(E, 0393) ~

^{T to a}

constant space

XT

^are commutative

diagrams

with _~

being

^a continuous function w.r.t. both

topologies.

^{With such} a ~

’P’

we can associate ^a function E - X which is continuous w.r.t. the

topol-

E

ogy r on E

by composing

~ with the evaluation-function

XT ~ X

^defined

by ~([f]t, t&#x3E;) = f(t).

^{It is not difficult to see that this}

correspondence

is bijective.

If we let U be the

forgetful

functor from

SPSP(T)

^{to the}

category

TOP

(of (external) topological

spaces and continuous

functions),

^which

p

associates to an internal space

(E, 0393) ~

^T the external space

(E, 0393)

which has E as its set of

points

^{and r as its}

topology,

^{and if we let C:}

TOP -

SPSP(T)

^{be the}

constant-space embedding

^{X ~}

XT,

^{then the}

correspondence

cp ^H

~’

amounts to

part (a)

^{of the}

following

theorem. Part

(b)

^is

proved similarly.

^{Here S:}

(TOP 1 T) - SPSP(T)

is the space of sections-functor

( Y - T) ~

gT.

2.3. THEOREM:

(a)

^{TOP is}

equivalent

^{to a}

reflective subcategory of SPSP(T),

^{and we have an}

adjunction U-|C.

(b)

^{we have an}

adjunction U’-|S,

^{where U’ is} the obvious

forgetful

p p p

functor (( E, 0393) ~ T ) H ( U(( E, 0393) ~ T) ~ ^{T ).}

In

fact,

connections between TOP and

SPSP(T)

^as

expressed

^{in this}

theorem ^can be formulated somewhat more

generally, by taking yet

another look at internal spaces. ^In order to make

things work,

^{we will for}

the remainder of this section assume that the base _{space T} is

always

^sober.

Recall

(cf. [1])

that if 03A9 is a locale

(cHa, frame),

^a

point

^or

superfilter

of 9 is a subset F c 9 with ~ ~

F,

^{T E}

F,

^{U A V E F - U E F and V E F}

(i.e.

^{F is a}

filter)

^{and moreover, for}

any d

^c

9,

VA c- ^{F a 3 U E} dUE F.

By pt( Q )

^we denote the space of

points

^{of 2} with the canonical

topology.

If E ⁼

((E, 0393) ^T)

^{is an} internal space in

Sh(T),

^we

get

^{a continu-}

j

ous function E

pte r)

^{from the}

embedding

^{r -}

(9(E).

2.4. LEMMA: _{Let p}

and j

^{be as above. In the}

diagram

there exists a

unique factorization

^r

of

p

through j.

PROOF:

Since j-1:

^{r -}

^O(E)

^is

injective,

^{i.e. a}

monomorphism

^of

locales, j

^{is an}

epimorphism

of sober spaces, ^so

uniqueness

of r is evident. As for

its

existence,

^{define for a}

superfilter

^{x E}

pte f),

It is _easy to see that

r*(x)

^{is a}

superfilter

ⁱⁿ

O(T),

^and

consequently,

there is ^a

unique point r(x) ~

^{T such that} _r

r*(x)={W~O(T)|r(x)~

W}.

This defines a function

pt(0393) ~ T,

^{which is}

continuous,

^since

r-1(W)={x~pt(0393)|r(x)~W}={x~pt(0393)|p-1(W)~x}=p-1(W)

~ 0393.

Finally, r o j = p,

^for

r( j(e)) = r({0 ~0393|e~0}={W|p-1(W)~

e} = p(e) (identifying

^real

points

^of sober spaces and

superfilters).

^D

p

2.5. LEMMA:

If f is

^{an internal} continuous

function from ( E, 0393) ~

^{T to}

( F, 0394) T,

then there exists a

unique

continuous

function

g

making

^the

diagram

^{below commute.}

PROOF: The

proof

is similar to that of 2.4. 0 Let SOB denote the

category

of sober spaces and continuous func- tions. The

preceding

^{two lemmas} then tell ^us that

TOP(Sh(T))

^is

equivalent

^{to a}

subcategory

^of

(SOB 1 T) - . (Note

^{that when E - T is a}

local

homeomorphism,

^{and T is}

sober,

^{so is}

E.)

2.6. COROLLARY:

TOP(Sh(T)) is equivalent

^{to the}

full subcategory of (SOB! T)- consisting of commuting triangles

with _p a local

homeomorphism, and j

^a

function

^which separates opens

( i. e.

if ^{U,’ V}

^are open subsets

of ^X,

^{then U =}

V ~ j(E) ~

^U

= j(E)

^~

V).

^D

Conversely,

every

object

^of

(SOB 1 T) ~

^{defines an} internal space, ^as in the

proof

^{of the}

following

^theorem.

2.7. THEOREM:

TOP(Sh(T))

^is

equivalent

^{to a}

coreflective subcategory of (SOB~T)~.

PROOF: The coreflector C maps ^a

triangle

on the internal space

(E, r ) 1 ^T,

^where

^{E 1}

^T is the local homeomor-

phism representing

the sheaf of sections of _p, and

0393 ~ O(E)

^{is the}

topology

determined

byf-1(O(Y))={f-1(U)U~O(Y)}.

^{The counit}

EA

CA ~ A of the

adjunction

^{is the}

isomorphisme

of sections

which is

easily

^{seen to factor}

through pt(0393) ~

^{Y. D}

3. Internal sober _spaces

Recall

(cf.

^for

example [1])

^{that a}

topological space X

^is called sober if _every

superfilter

^on

O(X)

^is

principal (i.e.,

is of the form

(O ~ O(X)|x ~ O}

^{for some}

^{x ~ X).}

^In this section we will _prove Fourman’s

representation

theorem for internal sober _spaces

(cf. [1], §8).

This is

straightforward, using

^the

representations

^of internal spaces discussed above.

Again,

^{we will}

throughout

this section assume that the base space ^T is sober.

Let us first look at what

superfilters

of opens ^are in an internal space

(E, 0393) ~

^{T. Let}

f

^{be a}

predicate

^on opens, ^or

equivalently,

^{an exten-}

sional function

r à (9(T) (extensional

^{here means that if}

0,0’e

^{r and}

p-1(W) ~ O = p-1(W) ~ O’, then W ~ f(O) = W nf( 0’),

^{for each W E}

O(T)). Trivially,

^{we can} calculate that

(1)

^and

(2)

^are

equivalent,

^for

U E

O(T).

(1) U ~ [f is a superfilter]

(2) f

satisfies the

following

three conditions

(in

^{addition to}

being extensional)

(i) U ~ f(E )

(ii) U~f(G)~f(H)= U~f(G~H), for G, H ~ 0393 (iii)

^{for each}

predicate

^{u on}

r,

i.e. extensional u : r -

(9(T),

U ~ f (~O~0393(O~ p-1u(O)))~~O~0393(u(O)~f(O)).

(Note

that from

(ii)

^{it follows that in}

(iii)

^we may

equivalently require

that

U~f(UO~0393(O~p-1(O))) = U~UO~0393(u(O)~f(O)).)

f ^-nu

3.1. LEMMA: Let

fU: 0393 ~ O(U)

^{be the}

composite 0393 ~ O(T) ~ O(U).

Then condition

(2)

^{above is}

equivalent

^to

fu being

^an

^V-map

^{such that}

fUo p-1 = idO(U).

PROOF:

(~)

Assume the

properties

ⁱⁿ

(2). Since f is extensional,

^we

get

that

W ~ fU p-1(W)

^for every W ~ U. And

conversely, applying (iii)

^to

the

predicate

^{u : 0393 ~}

O(T), u(O)=p(O)~ W,

it is clear

that fUp-1(W)

c W.

Thus fUp-1

⁼

_idO(U).

^From

⁽ⁱ⁾

^and

(ii)

it is clear

that fU

^{preserves T}

and A.

Also,

^if

~ r,

^{we can show}

that fU(U) = U{fU(G)|G ~ }

^as

follows.

(Of

course, ~ is

clear). By applying (iii)

^{to the}

predicate

u: 0393 ~

(9 (U)

^defined

by

we derive that

By extensionality

^of

f,

^the left hand side of this

equality

^is

equal

^to

fU(U),

^{and the}

ri ht

hand side is

equal

^to

U{fU1(G)|G}. ^(~)

Suppose fuis

^an

A -map

^{such that}

^{fu o p-1}

^{= id.}

Evidently, f is

^exten-

sional,

^and satisfies conditions

(i)

^and

(ii)

^of

(2).

^For

(iii),

^{U n}

fU(~O~0393(O~p-1u(O)))= U~UO~0393(fU(O)~fU(p-1u(O)))=

^U~

u(O)),

^for any

predicate

^{u : r -}

O(T).

^D

3.2. COROLLARY: An internal _space

(cf. Corollary 2.6)

is

internally

^sober

iff j

^{induces an}

isomorphism from

^sections

of

p ^to sections

of q (by a ~ j o a).

PROOF: If

( E, 0393)

^{T is}

internally ^sober,

sections of _p over U E

(9(T) correspond

^to

~V-maps f * :

^{r -}

O(U)

^{such that}

f* 0 p -’

⁼

idO(U), ^by

Lemma 3.1

(s:

^{U - E}

corresponds

^to

^s-1:

^{r -}

(9 (U)

^of

course).

^But

A V-maps

^{r -} f*

^{0 (U)}

^with

f * ° p-1

^{= id}

correspond by duality

^{to con-}

tinuous

sections f : U ~ pt(0393) satisfying ~W~O(T)(p-1(W)~f(t)~

t E

W),

^which

precisely

says that

f

^{is a} section of _q

(by

^the definition of _q

as in

2.4).

⁰

3.3. COROLLARY:

(cf. [1], §8). Every

internal sober _{space in}

Sh(T)

^{is a}

g

space

of

^sections gT

for

^some map

Y 9

^T

_of

sober _spaces. ~ 4. Pseudometric and metric spaces in sheaves

Topological space-objects

in sheaves ^are not as

nicely

^{behaved as}

alge-

braic

objects,

like commutative

rings.

^{In the}

algebraic

^{case one} may

regard

^{an internal}

ring-object,

^for

example, equivalently

^{as a sheaf of}

rings,

^{or as a sheaf of sets with a} continuous

ring-structure

^{on the stalks}

(cf.

^the

Introduction).

What remains of the first

correspondence

^{will be}

discussed in Part 2. The second

correspondence

^{has no}

topological analog:

in fact it is

quite

easy to construct

examples

which show that

(contrary

^{to the}

algebraic situation)

^the

topology

^{r of an} internal space

P

(E, 0393) ~

^{T is not} determined

by

^the

subspace topologies

it induces on

the

stalks p-1(t),

^{t ~ T. In this}

^section,

^{which is meant to}

provide

^some

more

examples

of internal

topological

spaces, it will be seen that

" being

determined

by

^{the structure on} the stalks" is

regained

^{if one considers a}

not-strictly topological

^structure like that of a

pseudometric

space.

Recall that the real

numbers-object

ⁱⁿ

Sh ( T )

^{is the constant}

space R

T

(cf. [1],

^for

example). Hence, by

^the

previous results, sheaf-maps f from

^a

sheaf A over T to

OB T correspond

^to continuous functions

f : EA - ^R,

where

EA

^{T is the}

étale-space representing

^A.

Henceforth,

^{the real}

numbers-object

^{in a} sheaf will

always

be denoted

by "811",

while "R"

refers to the external reals.

Using

^the

interpretation

^of

higher

^order

logic [1],

^{it makes sense to define a}

(pseudo-)metric

^as in the classical ^case:

4.1. DEFINITION: Let A be a sheaf over T. A

sheaf-map

^{d: A X A ~ is a}

pseudometric

^{on A if}

are all valid

(quantification

^{is over} "elements" of

A,

i.e. sections of the

representing

^local

homeomorphism EA ^T).

^{d is called a metric if in}

addition

The

following proposition

says what

pseudometrics

^{are from an exter-}

nal

point

^{of view.}

4.2. PROPOSITION: A

sheaf

map d: A X A ~ is a

pseudometric iff

^the

corresponding function

^d:

EA EA -

^{R is}

externally

^a continuous

pseudo-

metric on

fibers.

(Explanation.

Recall that the

correspondence

of continuous functions

EB

^{X and}

sheaf-maps f :

^{B ~}

XT,

^{for a sheaf B and a} space

X,

^is

given through

where

U,

^{is a} small nbd of e e

Es,

^{and a is a} section of

E, p

^T.

If Y - T is a continuous

function,

^a function d: Y ^X

TY ~

^{R is called}

a continuous

pseudometric

^on fibers if d is

continuous,

and for all

t ~ T,

the restriction

of d to p-1(t) p-1(t) is a pseudometric on p-1(t).)

PROOF: d

corresponds to d,

i.e. for sections _a, b:

U ~ EA of p,

^{and t E}

0//, d(a, b)(t) = d(a(t), b(t)). Clearly,

^{if d is a}

pseudometric

^on

^{fibers, d}

^is

an internal

pseudometric. Conversely,

^{if d is an internal}

pseudometric,

^we

find that d is a

pseudometric

^{on fibers}

by considering,

^for

points

^e,

e’ E EA with p(e)=p(e’),

the sections

(p Ue)-1, (p Ue’)-1,

^where

Ue, Ue.

^are small nbds of e and e’ such

that p(Ue) = p(Ue’).

⁰

Being

^{a metric is not a} "fiberwise"

property:

4.3. PROPOSITION: A

sheaf

map d : A X A ~ is a metric

(internally) iff

the

corresponding function

^d:

EA

^X

TEA - R

^{is a} continuous

pseudometric

on

fibers

^{with the} additional property ^Int

d-1({0}) ^{~ 0394} (d

⁼

0394E

^{is the}

diagonal {e, e&#x3E;|e~E}~E ETE).

PROOF:

By

5.2 it suffices to show that

( ~) If (e, e’)

^{E E X}

TE

^{is in Int}

d-1({0}),

then there are small nbds

Ue

of e and

Ue,

^{of e’ such that}

p ( Ue )

⁼

p(Ue’)

^and

Ue X T Ue.

^ç

d-1(0).

^Thus

for all

t ~ p(Ue) = p(Ue’), d((p ^P Ue) -1, ^(p rUe, ) -1 1)(t) = d((pUe) 1(t),

(p Ue’)-1(t))=0, so p(Ue)~[p Ue)-1 = (p Ue)-1].

( ~ )

Take sections a and b over U E

f9(T),

^{and take an} open ^W c

[d(a, b) = 0] = Int({t ~ U 1 d(a (t), b(t)=0}).

Then for all t E

W,

(a(t), b(t)&#x3E;

^{E Int}

d-l(O),

^so

a(t) = b(t).

^Hence W ç

[a

⁼

bl.

^1:1

If A is a sheaf on

T,

^{and d: A X A ~ is a}

pseudometric,

^{the internal}

topology

rd ^on d is defined

by setting

^for

global predicates

^{0 E}

(9(EA)

^on

T,

as in the classical case. Since the usual

proof

that this defines a

topology

is

contructively valid,

^we

get that Td

^{is indeed an internal}

topology.

^What

is its external

representation?

^{Let U be a small} open subset of

EA,

^{and let}

E:

p(U)~R+=(0,~)

^{be a} continuous function. The continuous bail around U of radius E,

B ( U, e),

^{is the set of}

points e ~ p-1 p(U)

^{such that}

3e’

E p-’p(e)

^~

Ud(e, e’) ~(p(e)).

P

4.4. PROPOSITION: Let a be a

sheaf

^on

T, EA ~

^{T its}

representation.

^{Let d}

be an internal

pseudometric

^{on A}

represented by

^d:

EA

^X

TEA -

^{R. Then}

the internal

topology

^induced

by

^d

corresponds

^to

(in

^{the sense}

of 1.5)

^the

topology

^on

EA having

^{the set}

of

continuous balls

B( U, ~),

^{with U small and}

p (U) 1 R+ continuous,

^{as a basis.}

PROOF: We have to show that the

global

internal opens ^are

exactly

^those

generated by

the basis defined in the

proposition.

^Let

O ~ O(EA). By definition,

^{0 E}

Frd

^iff

(1)

for all sections

U EA of p

and for all t e U ~

a-1(O)

there exist

a

nbd W ç

^{U and a} continuous E:

Wt~R+

such that for all sections b:

Wt ~ W ~ EA, {t ~ W|d(a(t), b(t)) ~(t)} ~ b-1(O).

We will show that

(1)

^is

equivalent

^to

(2)

^{Ve G O 3}

small nbd Ue of e ~~: Ue ~ R+: B(U, E)

^{c 0.}

(1) - (2).

Take e E O.

Using (1)

^{we find a nbd}

Wp(e)

^{and a continuous}

E &#x3E; 0

satisfying

the condition in

(1).

^{Now let}

U,

^{be a} small nbd of e with

p(Ue) ~ Wp(e),

^{and let}

~’ = ~p(Ue ).

^Then

B(Ue,~’) ~ O.

^{For if e’ e}

p-1p(Ue)

^with

e’ ~ B(Ue, ~’),

^{take a small nbd}

Ue.

of e’ such that

p(Ue’)~(Ue),

^and consider the section

(p Ue. ) -1.

Since e’ E

B ( Ue, ~’), d(e’,e")~(p(e’))

^{for some}

e" ~ p-1p(e’) ~ Ue, so p (e’) t 1 d( p

Ue’)-1(t), ^{( P P} Ue)-1(t)) ~(t)} ~ (p Ue’)(O),

^{so e’ E O.}

(2)

^~

(1).

^{Take a section}

U _{EA of p,}

^{and choose}

_{to E}

^Un

_a-1(O).

^Let

Ua(t0)

^{be a} small nbd of

a(t0),

^{and let E :}

p(Ua(t0))~R+

be such that

B(U, E) ç

⁰

(by (2)).

^{Then if}

^b: W ~ EA

^is

another

section of _p, with

W ~ p(Ue), d(a(t), b(t)) ~(t) implies b(t) ~

⁰

(for

^{t ~}

W), by

^defini-

tion of B (U, ~).

^D

PART 2. SPACES IN SHEAVES AND SHEAVES ^OF SPACES

In this

part,

^{we will}

point

^{out some} connections between internal

topological

spaces

(E, 0393)

^T and TOP-valued sheaves which have ^as their

underlying

^{sheaf of sets} the sheaf of sections

of E 1

^{T. As} has been remarked

before,

these connections are not as nice ^as in the case of internal

algebraic objects.

^{The main reason for this}

is,

^{of course, that the}

theory

^of

topology

^{is not a}

geometric theory,

^{it is not even a} first-order

theory.

The fact that the structure of internal spaces is ^not determined

by

the stalks is related to this. Another fact that makes

things

^{look not} very

hopeful

^{is that} TOP-valued sheaves are rather awkward

objects.

^{For if F}

is a TOP-valued sheaf on

T,

the space

FU,

^{U E}

O(T),

^{must be}

toplogi- cally

^embedded

FU -

03A0iFUi

in the

product Il i FU,

^for every

cover ( U, 1,

^of

^U,

^{as follows}

immediately

from the definition as

given,

^for

example,

ⁱⁿ

[3].

In the first section of this

part,

^we will prove ^a

general adjunction

theorem

relating

^the

category TOP( Sh ( T ))

^{of internal}

topological

^spaces

in

Sh(T)

^{and the}

category Sh ( T, TOP)

of sheaves on T with values in the

category

^{TOP of}

topological

spaces. In the second

section,

^{we will}

consider the

special

^{case of a}

locally compact

zero-dimensional

base-space.

Such spaces ^are

precisely

^the

Stone-spaces

of Boolean

rings (not

^neces-

sarily having

^a

unit).

1. A

general adjunction

^theorem

Let T denote the base space. We consider

assignments

B :

U ~ BU

of a set of subsets

Bu

^{of U to} each open U ^c

T,

^{such that}

(i) (monotone)

^{For U and V} open subsets of

T,

and K an

arbitrary

subset of

T,

^{if K c} V ~ ^U then

K ~ BV

^{iff K E}

BU.

(ii) (compact)

^{If K ~}

Bu

^and

( Ui 1,

^{is an open cover of}

^U,

^{then there}

are

finitely many Ui1, ... , Uin

^{and sets}

n

From now on, ^we will

arbitrarily

^{fix such a monotone and}

compact

assignment

^{B. Relative to this}

assignment B,

^we may then define ^a

functor

P

as follows. For an internal

space A

⁼

((EA, 0393) ~ T )

^{we define a sheaf}

F(B)(A)

^{with the same}

underlying

set-valued sheaf as A

(that is,

^sections

of p):

^{on each}

A(U) = {sections of p

^over

U},

^{U E}

O(T),

^{we define a}

topology by taking

^as subbasic opens the ^sets

for

K E BU

and 0 e r. Let us

verify

that this is well-defined:

PROOF:

(a)

^The

monotonicity

^condition

(i)

above makes restrictions

03C1=03C1UV: A(U) ~ A(V),

^for

^{U ~ V,}

continuous.

Also,

^if

U = UiUi,

^the

canonical inclusion

is ^a

topological embedding:

^we

only

^{have to} show that the

image

^{of a}

sub-basic open

[K, O] in A(U)

is open in

g(A(U)).

^But

by

^the

compact-

ness

property (ii) above,

^if

(bi)i=g(a)~g([K, O]).

then there are

UiJ, and KiJ ~ BUiJ,j=1,...,n, such that K=Kl1

^U

... ~Kin, so bij ~[Kij, O ^]

for j

⁼

1, ... , n,

^and

showing

^that

g([K, 0])

is open in

g( A ( U )).

(b)

^If

((EA, 0393) T) ^((EB, 0394) ^{T )}

^{is an}

( internal global)

^continu-

ous map, i.e.

qf = p, and f is

continuous w.r.t. both

topologies,

^{then each}

component of f as

^a natural transformation

f :

^{FA FB} is continuous:

this is

obvious, because f-1U ([K, O])

⁼

[ K, f-l( 0)].

⁰

Going

in the other

direction,

suppose ^{we are}

given

^a TOP-valued sheaf A with

underlying étale-space EA ^T,

^{and define a} collection

rA

of open

subsets of

EA by setting

^{for O E}

O(EA),

We then have

p

1.2. LEMMA:

(a) rA is

^a

p-topology

^on

EA -

^T

( b )

^The

assignment

^{A -}

((EA, 0393A) ^{T) defines}

^a

^functor

L(B): Sh(T, TOP) - TOP(Sh(T)).

PROOF:

(a)

^If

rA

^{is a}

topology,

^it

obviously makes p

continuous. So let’s show it is ^a

topology. Clearly, 0

^and

EA

^{are in}

0393A. ^Also,

^if

^{0,0’ E} rA,

then

O ~ O’ ~ 0393A,

^since

[K, O ~ O’] = [ K, O] ~ [ K, O’ ].

^For

unions,

suppose

O;

^E

FA

^for

each i E l,

and take U E

O(T),

^{K E}

Bu.

^If

^a(K)

^ç

U,O;,

^then

K ~~ia-1(Oi),

^{so there are}

K1 ~ Ba-1(Ot1),...,Kn ~ Ba-1(Otn)

such that

Ki

^{U ...}

~ Kn ^{= K,}

^and

thus a ~ ~nj=1[Kj, Oij]

^ç

^{[ K, UiOi ). This}

shows that

UiOi ~ rA

^whenever each of the

Oi e rA.

(b) L(B)

^{is a}

^functor,

^{for if}

^{f :}

^{A B and all}

components fu

^are

continuous,

^then

f

^is continuous as a map

Lq(B) (A) ~ L(B)(B) :

^write

L(B)(A)

⁼

^(EA, 0393A) ^T, L(B)(B)

⁼

^(EB, 0393B) ^T,

and take O ~

fB,

^i.e.

~U~O(T)~K~BU]K, O] is

open in

B ( U ). Then f-1(O)

also has this

property,

since for such U and

K,

{a~A(U)|a(K)~f-1(O)}=f-1U({b~B(U)|b(K)~O}).

1.3. THEOREM:

L(B) is left-adjoint

^to

F(B).

^D

PROOF: Let TJA:

A F(B)L(B)(A)

^{be the}

^identity,

for each TOP-valued sheaf A. The

components (~A)U

^{are all}

continuous,

^{as is}

easily

^verified.

To prove the

adjunction,

^{take a} TOP-valued sheaf

A,

^an internal space X ⁼

(E, 0394) ^T,

^{and a natural}

transformation f : A L(B)(X),

^{such that}

each

component fU is

continuous.

It suffices to show

that f is

continuous as a

morphism L(B) (A) ~ X.

^So

take O E à - ^we have to show that

f-1(O)

is open in

L(B)(A).

^{To this}

end, pick

^{U E}

O(T)

^{and K E}

Bu.

^Then

[K,f-1(O)] = f-1U([K, O ]),

^which